Heat the oil in a soup pot over medium heat. Add the onion and garlic and cook, stirring occasionally, until the onion is translucent, 4-6 minutes.Add the leek, carrots, and celery and cook, stirring occasionally, until softened, 5-7 minutes.

Add the tomato paste, stir well, and cook until it darkens, about 5 minutes.Add the broth, lentils, wine, vinegar, lemon, and sachet and bring to a simmer. Cook, uncovered, until the lentils are tender, about 40 minutes. Remove and discard the sachet and lemon half.